Properties

Label 2-2160-1.1-c3-0-58
Degree $2$
Conductor $2160$
Sign $-1$
Analytic cond. $127.444$
Root an. cond. $11.2891$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5·5-s − 11.2·7-s + 61.3·11-s − 75.7·13-s − 11.1·17-s − 71.5·19-s + 126.·23-s + 25·25-s + 235.·29-s − 110.·31-s + 56.4·35-s + 434.·37-s + 1.15·41-s + 77.6·43-s − 231.·47-s − 215.·49-s − 500.·53-s − 306.·55-s − 334.·59-s + 147.·61-s + 378.·65-s − 84.9·67-s − 101.·71-s − 50.4·73-s − 693.·77-s + 818.·79-s + 206.·83-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.610·7-s + 1.68·11-s − 1.61·13-s − 0.159·17-s − 0.863·19-s + 1.14·23-s + 0.200·25-s + 1.50·29-s − 0.642·31-s + 0.272·35-s + 1.92·37-s + 0.00441·41-s + 0.275·43-s − 0.718·47-s − 0.627·49-s − 1.29·53-s − 0.752·55-s − 0.738·59-s + 0.308·61-s + 0.722·65-s − 0.154·67-s − 0.169·71-s − 0.0809·73-s − 1.02·77-s + 1.16·79-s + 0.272·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2160\)    =    \(2^{4} \cdot 3^{3} \cdot 5\)
Sign: $-1$
Analytic conductor: \(127.444\)
Root analytic conductor: \(11.2891\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{2160} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 2160,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 + 5T \)
good7 \( 1 + 11.2T + 343T^{2} \)
11 \( 1 - 61.3T + 1.33e3T^{2} \)
13 \( 1 + 75.7T + 2.19e3T^{2} \)
17 \( 1 + 11.1T + 4.91e3T^{2} \)
19 \( 1 + 71.5T + 6.85e3T^{2} \)
23 \( 1 - 126.T + 1.21e4T^{2} \)
29 \( 1 - 235.T + 2.43e4T^{2} \)
31 \( 1 + 110.T + 2.97e4T^{2} \)
37 \( 1 - 434.T + 5.06e4T^{2} \)
41 \( 1 - 1.15T + 6.89e4T^{2} \)
43 \( 1 - 77.6T + 7.95e4T^{2} \)
47 \( 1 + 231.T + 1.03e5T^{2} \)
53 \( 1 + 500.T + 1.48e5T^{2} \)
59 \( 1 + 334.T + 2.05e5T^{2} \)
61 \( 1 - 147.T + 2.26e5T^{2} \)
67 \( 1 + 84.9T + 3.00e5T^{2} \)
71 \( 1 + 101.T + 3.57e5T^{2} \)
73 \( 1 + 50.4T + 3.89e5T^{2} \)
79 \( 1 - 818.T + 4.93e5T^{2} \)
83 \( 1 - 206.T + 5.71e5T^{2} \)
89 \( 1 - 648.T + 7.04e5T^{2} \)
97 \( 1 - 1.88e3T + 9.12e5T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.351614266176739854848539253283, −7.48645245781778845626125764384, −6.67013207047215842814041817779, −6.28437737909305034887817509677, −4.89345235458928274085556640474, −4.34088185540242154088978749853, −3.33854041190009849974516095342, −2.45109460755093593343389066785, −1.13885379776093629722219097671, 0, 1.13885379776093629722219097671, 2.45109460755093593343389066785, 3.33854041190009849974516095342, 4.34088185540242154088978749853, 4.89345235458928274085556640474, 6.28437737909305034887817509677, 6.67013207047215842814041817779, 7.48645245781778845626125764384, 8.351614266176739854848539253283

Graph of the $Z$-function along the critical line